evaluated at \(z=0\) . The second formula () is a consequence of the definition of the inverse of the Riemann–Hilbert problem (REF ), which can be found in formula (2.16) of [1]}. Indeed,
\({Y}^{(-n)}(s;z) = \left( \begin{matrix} -2\pi i \mathcal {C}({W}{\hat{P}}_{n-1}^*)(s;z){\gamma }_{n-1}(s) & -\mathcal {C}({W}{\hat{P}}_n^*)(s;z)\\2\pi i {\hat{P}}_{n-1}^*(s;z){\gamma }_{n-1}(s) & {\hat{P}}_n^*(s;z)\end{matrix}\right).\) 
